Tangent bundle of $S^1$ is diffeomorphic to the cylinder $S^1\times\Bbb{R}$

We can parametrise the circle $\mathbb S^1 \subset \mathbb R^2$ by $\theta \mapsto (\cos\theta,\sin\theta)$.

Any vector tangent to the circle at a point must be perpendicular to the position vector of that point. (This is a classical theorem: the radius is perpendicular to the tangent line.)

The tangent vectors are of the form $\lambda \mapsto (-\lambda\sin\theta,\lambda\cos\theta)$, for some $\lambda \in \mathbb R$.

The Tangent bundle to the circle can be parametrised by $$(\theta,\lambda) \longmapsto (\cos\theta,\sin\theta,-\lambda\sin\theta,\lambda\cos\theta).$$

If you like to be formal then you can write this as $$\mathrm f : (\theta,\lambda) \longmapsto (\cos\theta,\sin\theta;[-\lambda\sin\theta]\partial_x+[\lambda\cos\theta]\partial_y)$$

where $[-\lambda\sin\theta]\partial_x+[\lambda\cos\theta]\partial_y$ is a member of $T_p\mathbb R^2$, and $p = (\cos\theta,\sin\theta)$.

The cylinder appears because $\mathrm f$ is $2\pi$-periodic in $\theta$.

A tangent vector to a circle is not really $(v_1,v_2)$. Rather, it is of the form $c\frac{d}{d\theta}$ where $c\in\mathbb{R}$ is a real number and $\frac{d}{d\theta}$ is the standard vector field globally defined on the whole circle. Therefore you can send $(x,v)$ to $(x,c)\in S^1\times\mathbb{R}$.

As you know $(x_2,-x_1)\in T_xS^1$. So let $\phi:T_xS^1\to S_1\times\mathbb{R}$ given by $$\phi(x_1,x_2,v_1,v_2)=(x_1,x_2,<(x_2,-x_1),(v_1,v_2)>)=(x_1,x_2,v_1x_2-v_2x_1).$$ So $\phi^{-1}(x_1,x_2,t)=(x_1,x_2,t(x_2,-x_1))=(x_1,x_2,tx_2,-tx_1)$.

For a conceptual understanding as well, imagine you have a circle in the xy-plane centered at the origin. At each point on the circle, the tangent space "looks like" a line in the xy-plane tangent to that point. Now rotate this line until it becomes perpendicular to the xy-plane. Do this for all points on the circle. What we get is a cylinder!